He did it very well, did David Cameron. One of the dimensions of being prime minister at which he excels is crafting the right language and striking the appropriate tone on grave or shocking occasions or, in this case, responding on behalf of both government and country to a shockingly grave report. He delivered a model statement of penitence for what he correctly called "the double injustice" done to the victims of the Hillsborough stadium crush.

It is always easier, mind, to say sorry for a disaster that was someone else's fault. The bigger test is what happens next. It is wrong to think of Hillsborough, and the disgusting conduct of some members of the South Yorkshire force, as a tragedy to be deeply regretted and then filed away as an event belonging to the distant past. It is true that stadium design has been massively improved and methods of crowd control have become more sophisticated. Football hooliganism, fear of which was a contributory factor, has largely disappeared from Britain. So a tragedy of that type is less likely to happen now.

What cannot be said with so much confidence is that the police would never behave like that again. It would be a comfort to think that Hillsborough was an isolated example from long ago of officers attempting to conceal their own neglect and incompetence by lying about what had happened and smearing the victims of their mistakes. It would also be a false consolation. What happened at Hillsborough was not an isolated one-off, but a particularly horrific example from a pattern of policing scandals over many decades. The very long list includes multiple miscarriages of justice such as the false imprisonment of the Birmingham Six and the Guildford Four. The shooting of Jean Charles de Menezes and the death of Ian Tomlinson at the G20 protests are two more recent cases where the police made appalling errors and then tried to cover up their mistakes. Scotland Yard's failure to properly investigate the hacking scandal exposed alarming lacunae in the ethical and professional judgment of some of the most senior officers in the Met. Other officers face trials for corruptly selling information to newspapers.

There are many dedicated men and women doing a job which can be highly demanding, often dangerous, and sometimes life-threatening, and they do it in a spirit of public service. But that cannot be said of the police as a collective. Hillsborough is a particularly grotesque illustration of a culture that to this day too often puts its own interests before those of the public it is paid to serve. Repeated scandal is what happens when you have a closed profession highly resistant to scrutiny or criticism whose default instinct is to protect its own.

It is also the result of a failure of political will to pursue reform. The approach of Margaret Thatcher and Tony Blair was not to challenge police working practices or culture. For her, they were "one of us" and crucial allies in her epic battles with the trades unions. Mrs Thatcher responded to an earlier report about Hillsborough by telling her then home secretary, Douglas Hurd, to mute the government response to devastating evidence of police misconduct. For Tony Blair, being pro-police was a vital ingredient of being New Labour. He loaded them with many additional powers and generously increased budgets. To a reform-minded home secretary, Mr Blair once demanded: "Do we really want a fight with the police?"

To see what he was frightened of, you only have to look at the trade union of their rank-and-file officers, the Police Federation. Visiting the Federation's annual conference is the scariest date in the home secretary's diary. Earlier in the year, they boorishly humiliated Theresa May when she tried to make her voice heard through the booing and heckling. She should not take that personally. They dealt out the same treatment to Labour home secretaries too.

Millions of employees in the private sector have learnt that there is no longer such a thing as a "job for life" while millions in the public sector have been challenged to justify their existence. Working practices have been revolutionised; organisations radically restructured; pay and conditions dramatically changed. Teachers, nurses, ambulance drivers, doctors, local authority workers and firefighters have all felt bracing blasts of government-driven change. The big exception has been the police.

Because Labour prime ministers tend to become nervous of looking anti-police, it probably takes Conservative ministers to reform the forces of law and order. David Cameron and Theresa May made a bold start when they commissioned the first fundamental review of police pay and practices in 30 years by Tom Winsor, the former rail regulator. They have also demonstrated a seriousness of intent about implementing his recommendations by making him chief inspector of constabulary, an appointment which has provoked fierce outrage in the ranks. When he takes up the post this autumn, it will be the first time that the role has been performed by someone from outside the police since its creation in 1856.

The government has also set great store by the novelty of elected police and crime commissioners, the most dramatic change in police governance ever. Making chief constables accountable to commissioners, so it is believed, will challenge the whole culture of policing. Forces will have someone independent constantly looking over their shoulders. One minister who is an enthusiast for commissioners says: "What would have happened in South Yorkshire in 1989 had there been an elected person in charge? Their whole purpose in life would have been to find out what the hell happened."

I would certainly hope so, but there are problems with the commissioners even before a single one is in post. The quality of the candidates is patchy, ranging from people with the potential to be excellent invigilators to old party hacks. The number of people turning out to vote is going to be abysmal because, when the original timetable slipped, the government took a crazy decision to make these standalone elections in the inclement month of November rather than combine them with the county council elections next May. Most informed people I talk to, including those running the campaigns, reckon they will be lucky if turnout averages 15%. Commissioners elected on such a small vote may have legitimacy issues which will embolden the police to try to ignore them.

An even more important question is whether the Winsor reforms can be pushed through against the ferocious hostility of many officers. They don't like his proposal to widen the pool of talent from which police leaders are drawn by allowing direct entry into the higher ranks, breaking the tradition that all officers must start out as a beat constable. Another resisted reform is that officers should be rewarded for their skills, performance and hours spent on the front line rather than the length of service. Were we talking about any other part of the public sector, ministers could just get on and implement these changes. They can't do this with the police because of the no-strike deal. So the fate of Mr Winsor's most crucial recommendations are being determined by a body called the police arbitration tribunal.

On top of which, the minister most passionate about reform has just quit. Nick Herbert, who had a real zeal for the challenge, left the government in the recent reshuffle. He has been replaced by Damian Green, who was moved to the police brief from immigration, and must think that all the clichés about the frying pan and the fire apply. A few years back, Mr Green was the target of a ludicrously misconceived police operation when officers raided the MP's home and parliamentary offices. So Mr Green is probably not the sort to go soft on police reform. In the end, though, the outcome of this battle will most depend on the mettle of his seniors, the home secretary and the prime minister. The police never come quietly. As one former chief constable puts it, there is an "inbuilt conservatism" about the service which makes it "often risk-averse, process-dominated and defensive." It is a big thing for governments to take on the police, which is why ministers have so rarely done it.

David Cameron demonstrated how accomplished he can be at making a pitch-perfect parliamentary statement with his eloquent apology for Hillsborough. Getting the police to accept and implement reform will require different leadership skills. This will be a test of how much steel there is in his spine.